In recent years, the linear economic model and global warming have shown that it is necessary to move toward a circular economic model. In this scenario, the recycling of waste electrical and electronic equipment (WEEE) with green processes is one of the pending tasks; thus, in the present review, advances in the solid–liquid and liquid–liquid extraction processes, processes among the most important for the recovery of metals from ores or WEEE, with green solvents such as deep eutectic solvents (DES) are presented and analyzed, identifying the present and future challenges. To date, most articles focused on one of the processes, be it solid–liquid or liquid–liquid extraction, while few reports included the entire hydrometallurgical process, which could be due to heterogeneity of the WEEE, a characteristic that influences determining the leaching kinetic and the leaching mechanisms. A deeper understanding of the phenomenon would help improve this process and the next stage of liquid–liquid extraction. This also leads to the fact that, at the liquid–liquid extraction stage, most articles considered synthetic pregnant leach solutions to evaluate each of the variables, whereas the stripping of the ions and the recycling of the DESs in continuous processes is a challenge that should be addressed in future work. From the analysis, for WEEE leaching, it was identified that acid DESs are those achieving the best extraction percentages in the leaching of copper, lithium, and cobalt, among others, where the most studied hydrogen bond acceptor (HBA) is choline chloride with an acid (e.g., citric or lactic acid) as the hydrogen bond donor (HBD). For the liquid–liquid extraction of ions is a greater variety of HBAs (e.g., lidocaine, trioctylphosphine oxide and triphenyl phosphate) and HBDs (e.g., decanoic acid, thenoyltrifluoroacetone, and benzoyltrifluoroacetone) used; however, studies on the extraction of cobalt, lithium, copper, and nickel stand out, where the pH and temperature parameters have great influence.